.TH  ZPBSV 1 "November 2006" " LAPACK driver routine (version 3.1) " " LAPACK driver routine (version 3.1) " 
.SH NAME
ZPBSV - the solution to a complex system of linear equations  A * X = B,
.SH SYNOPSIS
.TP 18
SUBROUTINE ZPBSV(
UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
.TP 18
.ti +4
CHARACTER
UPLO
.TP 18
.ti +4
INTEGER
INFO, KD, LDAB, LDB, N, NRHS
.TP 18
.ti +4
COMPLEX*16
AB( LDAB, * ), B( LDB, * )
.SH PURPOSE
ZPBSV computes the solution to a complex system of linear equations
   A * X = B,
where A is an N-by-N Hermitian positive definite band matrix and X
and B are N-by-NRHS matrices.
.br

The Cholesky decomposition is used to factor A as
.br
   A = U**H * U,  if UPLO = \(aqU\(aq, or
.br
   A = L * L**H,  if UPLO = \(aqL\(aq,
.br
where U is an upper triangular band matrix, and L is a lower
triangular band matrix, with the same number of superdiagonals or
subdiagonals as A.  The factored form of A is then used to solve the
system of equations A * X = B.
.br

.SH ARGUMENTS
.TP 8
UPLO    (input) CHARACTER*1
= \(aqU\(aq:  Upper triangle of A is stored;
.br
= \(aqL\(aq:  Lower triangle of A is stored.
.TP 8
N       (input) INTEGER
The number of linear equations, i.e., the order of the
matrix A.  N >= 0.
.TP 8
KD      (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = \(aqU\(aq,
or the number of subdiagonals if UPLO = \(aqL\(aq.  KD >= 0.
.TP 8
NRHS    (input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B.  NRHS >= 0.
.TP 8
AB      (input/output) COMPLEX*16 array, dimension (LDAB,N)
On entry, the upper or lower triangle of the Hermitian band
matrix A, stored in the first KD+1 rows of the array.  The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = \(aqU\(aq, AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
if UPLO = \(aqL\(aq, AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
See below for further details.

On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**H*U or A = L*L**H of the band
matrix A, in the same storage format as A.
.TP 8
LDAB    (input) INTEGER
The leading dimension of the array AB.  LDAB >= KD+1.
.TP 8
B       (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if INFO = 0, the N-by-NRHS solution matrix X.
.TP 8
LDB     (input) INTEGER
The leading dimension of the array B.  LDB >= max(1,N).
.TP 8
INFO    (output) INTEGER
= 0:  successful exit
.br
< 0:  if INFO = -i, the i-th argument had an illegal value
.br
> 0:  if INFO = i, the leading minor of order i of A is not
positive definite, so the factorization could not be
completed, and the solution has not been computed.
.SH FURTHER DETAILS
The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = \(aqU\(aq:
.br

On entry:                       On exit:
.br

    *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
    *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
   a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

Similarly, if UPLO = \(aqL\(aq the format of A is as follows:
.br

On entry:                       On exit:
.br

   a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
   a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
   a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

Array elements marked * are not used by the routine.
.br

